for \(l=1,2\)  (cf. Eq. (REF )) consider \(\phi ^{\eta \chi }(p):= (|x|+|t|)^{-1}\big (px-t\eta \chi \,|p|\big )\) . Then \(\nabla \phi ^{\eta \chi }(p)= (|x|+|t|)^{-1}\big (x-\frac{t\eta \chi }{|p|}p\big )\)  and \(|\nabla \phi ^{\eta \chi }(p)|\ge (|x|+|t|)^{-1}(|t|-|x|)\)  for \(p\ne 0\) . Now suppose \(|x|\le v|t|\) . Then \(|\nabla \phi ^{\eta \chi }(p)|\ge (v|t|+|t|)^{-1}(|t|-v|t|)=\frac{1-v}{1+v}>0\) . This implies (cf. [1]}) for \(\eta ,\chi \in \lbrace +,-\rbrace \) , \(l=1,2\) , and for every \(N>0\)  that there is a finite constant \(A_N\)  with
\(\big |\big (\psi ^\eta _t(x)\big )_l\big |\le A_N(1+|x|+|t|)^{-N}\textrm { \;if\; } |x|\le v |t|\) 
